1. Use strong induction to show that if you can run one mile or two miles, and if you can always run two more miles once you have run a specified number of miles, then you can run any number of miles.

Proof

Step-by-step explanation:

Given:

- If you run 1 or 2 miles

- You can always run 2 miles more after running specified number of miles

Find:

- Prove that you can run any number of miles

Solution:

- Let M (n) be " You can run the nth mile"

Basis step: n = 1 and n = 2

- M (1) and M (2) are True, because you can run one or two miles as given in statement.

Inductive Step:

- We assume that M (1), M (2), ..., M (k) are all true, thus you can run the first k miles.

- We then need to prove that M (k + 1) is also true.

- Since M (k - 1) is true then M (k + 1) is true. (You can always run 2 miles more after running specified number of miles)

Conclusion:

- By the principle of strong induction, M (n) is true for all positive n integers.